Potts model coupled to causal triangulations

TL;DR

This paper investigates the Potts model coupled with causal triangulations on a torus, establishing duality relations and bounds for the free energy to understand phase transitions and Gibbs measure behavior.

Contribution

It introduces a duality relation for the Potts model on causal triangulations and delineates regions for the existence and uniqueness of Gibbs measures.

Findings

Derived duality relation between free energies of the model and its dual.

Identified regions where the Gibbs measure exists and is unique.

Provided bounds for the infinite-volume free energy.

Abstract

In this work we study the annealed Potts model coupled to two dimensional causal triangulations with periodic boundary condition. Using duality on a torus, we provide a relation between the free energy of the Potts model coupled CTs and its dual. This duality relation follows from the FK representation for the Potts model. In order to determine a region where the critical curve for the model can be located we use the duality relation and the high-temperature expansion. This is done by outlining a region where the infinite-volume Gibbs measure exists and is unique and a region where the finite-volume Gibbs mea- sure has no weak limit (in fact, does not exist if the volume is large enough). We also provide lower and upper bounds for the infinite- volume free energy.